sphere
volume of sphere = $\frac{4}{3}pi r^{3}$
volume of cylinder = $pi r^{2}h$
volume of torus = $2pi^{2}rr$
exercise 4e
1 the tip of the minute hand of a clock travels from 12 to 3. calculate the angle through which the hand moves, in degrees, revolutions and turns.
2 when the minute hand of a clock is on 2 and the hour hand is on 8, what is the angle between the two? give your answer in revolutions.
3 state the type of angle: 42°, 90°, 99°, 200°, 180° and 360°.
4 give an example of:
a two complementary angles
b two supplementary angles
5 show by means of labelled diagrams and actual values:
a adjacent angles
b vertically opposite angles
c alternate angles
d corresponding angles
e co - interior angles.
6 in the diagram, ab is parallel to cd and xy is a transversal. find the values of all the angles represented by letters.

Question

Accepted Answer

### 1.
# Explanation:
## Step1: Recall clock - angle relationship
A full - circle of a clock is 360°. The minute hand moves 360° in 60 minutes or 12 hours. When it moves from 12 to 3, it covers 3 out of 12 equal parts of the clock - face.
## Step2: Calculate angle in degrees
The angle in degrees is $\frac{3}{12}	imes360^{\circ}=90^{\circ}$.
## Step3: Calculate angle in revolutions
Since 1 revolution = 360°, the number of revolutions is $\frac{90^{\circ}}{360^{\circ}}=\frac{1}{4}$ revolutions.
## Step4: Calculate angle in turns
1 turn = 360°, so the number of turns is $\frac{1}{4}$ turns.
# Answer:
In degrees: 90°, In revolutions: $\frac{1}{4}$ revolutions, In turns: $\frac{1}{4}$ turns

### 2.
# Explanation:
## Step1: Recall the angular separation between hour - marks
The angle between two consecutive hour - marks on a clock is $\frac{360^{\circ}}{12}=30^{\circ}$.
## Step2: Calculate the angular separation between 2 and 8
The number of hour - marks between 2 and 8 is 6. So the angle in degrees is $6	imes30^{\circ}=180^{\circ}$.
## Step3: Convert to revolutions
Since 1 revolution = 360°, the number of revolutions is $\frac{180^{\circ}}{360^{\circ}}=\frac{1}{2}$ revolutions.
# Answer:
$\frac{1}{2}$ revolutions

### 3.
# Explanation:
## Step1: Recall angle - type definitions
An acute angle is between 0° and 90°, a right - angle is 90°, an obtuse angle is between 90° and 180°, a reflex angle is between 180° and 360°, and a full - angle is 360°.
## Step2: Classify the angles
$42^{\circ}$ is an acute angle, $90^{\circ}$ is a right - angle, $99^{\circ}$ is an obtuse angle, $200^{\circ}$ is a reflex angle, $180^{\circ}$ is a straight - angle, $360^{\circ}$ is a full - angle.
# Answer:
$42^{\circ}$: Acute angle, $90^{\circ}$: Right - angle, $99^{\circ}$: Obtuse angle, $200^{\circ}$: Reflex angle, $180^{\circ}$: Straight - angle, $360^{\circ}$: Full - angle

### 4.
# Explanation:
## Step1: Recall complementary and supplementary angle definitions
Complementary angles add up to 90° and supplementary angles add up to 180°.
## Step2: Provide examples
a) For complementary angles, 30° and 60° since $30^{\circ}+60^{\circ}=90^{\circ}$.
b) For supplementary angles, 120° and 60° since $120^{\circ}+60^{\circ}=180^{\circ}$.
# Answer:
a) 30° and 60°
b) 120° and 60°

### 5.
# Brief Explanations:
a) Adjacent angles share a common vertex and a common side. For example, in a right - angled triangle, the two non - right angles adjacent to the right - angle share the vertex of the right - angle and a common side.
b) Vertically opposite angles are formed when two lines intersect. If two lines $l_1$ and $l_2$ intersect at a point $O$, the angles opposite each other are equal. For example, if one angle is 50°, the vertically opposite angle is also 50°.
c) Alternate angles are formed when a transversal intersects two parallel lines. Alternate interior angles are equal. For example, if a transversal intersects two parallel lines, and one alternate interior angle is 40°, the other is also 40°.
d) Corresponding angles are formed when a transversal intersects two parallel lines. Corresponding angles are equal. For example, if a transversal intersects two parallel lines, and one corresponding angle is 60°, the other is also 60°.
e) Co - interior angles are formed when a transversal intersects two parallel lines. Co - interior angles add up to 180°. For example, if one co - interior angle is 110°, the other is 70°.
# Answer:
Diagrams can be drawn to illustrate these concepts with appropriate angle - value labels as described above.

### 6.
# Explanation:
## Step1: Recall angle - relationships for parallel lines and a transversal
Corresponding angles are equal, vertically opposite angles are equal, alternate interior angles are equal, and co - interior angles are supplementary.
## Step2: Identify angle relationships
Given an angle of 50°.
$x = 50^{\circ}$ (vertically opposite angles).
$p=50^{\circ}$ (corresponding angles with the 50° angle).
$w = 180^{\circ}-50^{\circ}=130^{\circ}$ (co - interior angle with the 50° angle).
$q = 50^{\circ}$ (alternate interior angle with the 50° angle).
$r = 130^{\circ}$ (vertically opposite to $w$).
$u = 50^{\circ}$ (vertically opposite to $q$).
$v = 130^{\circ}$ (vertically opposite to $r$).
# Answer:
$x = 50^{\circ}$, $p = 50^{\circ}$, $w = 130^{\circ}$, $q = 50^{\circ}$, $r = 130^{\circ}$, $u = 50^{\circ}$, $v = 130^{\circ}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1.

Step1: Recall clock - angle relationship

Step2: Calculate angle in degrees

Step3: Calculate angle in revolutions

Step4: Calculate angle in turns

Step1: Recall the angular separation between hour - marks

Step2: Calculate the angular separation between 2 and 8

Step3: Convert to revolutions

Step1: Recall angle - type definitions

Step2: Classify the angles

Answer:

2.